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Sets
Concept of Sets 1. A set is a collection or a group of any kind of objects. 2. There are two ways to define a set: (a) description (b) using set notations The definition of set has to be clear and precise so that the elements can be identified. The notation used for sets is braces, { }. 3. A set is usaually denoted by a capital letter. 4. The objects that make up a set are called elements or members of the set. 5. Hence, a set containing elements a, b, c and d is denoted by {a, b, c, d}. 6. If x is an element of set A, we write x ? A: conversely, if x is not an element of A, we write x ? A. 7. A Venn diagram is a pictorial presentation used to represent a set or more commonly used to illustrate the relationship between sets. 8. A circle is usually used to represent a given set but there is no hard and fast rule to say that it must be represented by a circle. A set can be represented by a traingle or any other shapes which isconvenient. The elements of the set are enclosed in the circle. 9. The notation n (A) denotes the number of elements in set A. 10. We use a 'dot' followed by the value of the element to represent a particular element whereas we use a number to represent the number of elements in a Venn diagram. 11. An empty set is a set which has no elements. An empty set is also called a null set. 12. We donate the empty set by the symbol ? (phi) or { }. The symbol ? (phi) is used more frequently. 13. Students must not be confused with {0} and ?. There is a difference between them. 14. {?} is a set with one element whose value is zero whereas ? is an empty / null set with no element in it. Hence, if A = {0} and B = ?. 15. {?} is also not an empty set because it is a set with one element which is ?. 16. Equal sets have identical elements, that is, they contain exactly the same elements. Subset, Universal Set and the Complement of a set 1. If every element of set A is also an element of set B, then A is a subset of B. We write "A ? B". 2. If A is noy a subset of B, we write "A ? B". 3. Every set is a subset of B, namely "A ? A". 4. The empty set is a subset of every set, namely "? ? A". 5. The number of possible subsets for set A is given by the formula 2 (power n) where n is the number of elements in set A. 6. The relationship between a set and its subset can be represented by a Venn diagram. 7. A universal set is a specified set which contains all the elements being discussed.